I assume that it is meant "by powder diffraction evaluating the diffraction line broadening". In this case you still should be sure that your entire microstrain broadening is due to microstrain from dislocations. Otherwise your dislocation densities constitute some upper bound for the dislocation density....

Dear Uday

Have a look at the appended link ... also you will find numerous contributions from M. Wilkens as far as I know the still up-to-date theory behind it.

As far as I know PM2K has implemented the Wilkens-model but honestly dislocation and deformation theory is not my field of expertise ...

Regards

http://dx.doi.org/10.1107/S002188980702078X

I assume that it is meant "by powder diffraction evaluating the diffraction line broadening". In this case you still should be sure that your entire microstrain broadening is due to microstrain from dislocations. Otherwise your dislocation densities constitute some upper bound for the dislocation density....

These are some potential effects of defects and strain fields on the Bragg profile, collected from various sources in the literature over many years. https://www.flickr.com/photos/85210325@N04/15649120475/ 

The WPPM approach for powder XRD data by Leoni, Scardi et al shared by Dominique is a sound approach. However, as Andreas points out, it is important to deconvolute the effects of various Nano structural contributors to the Bragg profile changes carefully when the measured XRD signal is a convolution of several of these factors. Especially, the "micro lattice strain effect" and the "particle size effect" (DDS-diffracting domain size).

For severely deformed materials, some sort of a W-H (Williamson-Hall) approach may work. This method may help deconvolute the two competing effects by examining the profiles of higher order reflections as well. This may not be trivial. I'm interested in the solution to this challenge!

Example 1: of the XRD signal (Laue transmission mode) from a highly deformed (plastic) aluminum foil with some amount of "recrystallization": https://www.flickr.com/photos/85210325@N04/7944785794/in/set-72157632728981912

Example 2: of a mono crystal of GaAs compared with the theoretical LEPTOS generated Bragg profile: https://www.flickr.com/photos/85210325@N04/9430820747/in/set-72157635172219571

It all comes down to the definitions ultimately. Sigmund Weissmann et al of Rutgers University have done considerable work in attempting to deconvolute various Nano structural effects on the Bragg reflections with poly crystalline materials. I'll post references when I find them.

Here is one relationship that has been used for dislocation density over decades for a Gaussian Bragg profile: Dislocation Density, ρ=β2/9b2 for a Gaussian distribution, b=Burger’s Vector, ρ-Dislocation Density,  integral breadth, β, is related to the FWHM peak width, H, by β=0.5H(π/loge2)1/2. If someone could find the original paper (1950's I think) referencing this relationship, I'd be much obliged. I misplaced it again:-)

Dear Uday ,

I send to you a book titled (Non-Destructive tests) in Arabic included a chapter about x-ray testing

Best Regards

I would recommend x-ray topography to get dislocation density (and types).

Here are some useful references:

Bowen, Keith and Tanner, Brian: High Resolution X-Ray Diffractometry and Topography. Taylor and Francis (1998).

It is even possible to get information on the 3D spatial distribution using topo-tomography:

W. Ludwig, P. Cloetens, J. Härtwig, J. Baruchel, B. Hamelin and P. Bastie: Three-dimensional imaging of crystal defects by `topo-tomography'. J. Appl. Cryst. (2001) 34, 602-607.

P. Mikulík, D. Lübbert, D. Korytár, P. Pernot, and T. Baumbach. Synchrotron area diffractometry as a tool for spatial high-resolution three-dimensional lattice misorientation mapping. J.Phys. D:Appl. Phys. (2003) 36(10), A74-A78.

The determination of dislocation densities in severely deformed copper (cold worked by rolling and/or filings) by NMR technique using the nuclear  quadruple line broadening theory developed by us was described in our Ph.D. thesis at Stanford 1964. This work was also published in METU Journal of Pure and Applied Sciences, Vol.1, No.2 (1968) pp 155-173. Where we found that dislocation densities are 1.8 x 10*11 cm-2 and 7.5 x 10*11 cm-2  in samples deformed by rolling and filings, respectively. Where we have employed our theoretical work on the analysis of Faulkner’s (1959) and Bloembergen and Rowland's (1953) experimental measurements of the derivative maximum Gmax and the integrated intensity I versus Gmax ratio for various deformations. Unfortunately, this work disappeared in shelves of METU's and Harward  libraries !! But extensive  used by   O. Kanert (2006) in his highly regarded work in Germany.Best Regards.

İn the  NMR data analysis the most important problem is the evaluation of the Block electron contribution to the line broadening in regards to the dislocation strain effects. One of our main contribution to this subject is the theoretical formulation and the numerical computation of Block enhancement factor using the tabulated atomic core  Cu+  wave functions from Hartree,  which is found to  be 7.62. 

The volume weighted average domain size (Dv) and the microstrains (ε) can be determine using Williamson–Hall relation . So the dislocation density was calculated through Rietveld method, the average dislocation density ρ can be estimated from the relation ρ = (ρd . ρs)1/2, where, ρd = 3/Dv 2 (dislocation density due to the domain), ρs = 〈ε^2 〉/b2 (dislocation density due to the microstructure), and b the Burger’s vector.

Please read those two references carefully and good luck 

G.K. Williamson, W.H. Hall. Acta Metall. 1(1953) 22–31.

G. K. Williamson, R. E. Smallman, Philos Mag. 1 (1956) 34–46.

Simple way to calculate the dislocation density

R=2*sqrt (3)*E/D*B=1.73*E/D*B

where  R  - is the dislocation density [m-2], E - microstresses lattice  [%], D - coherent scattering area [m], B -  Burgers vector. B=a/sqrt(2). a - lattice parameter [m]

1. Williamson G.K., Smallman R.E. // Phil.Mag. 1956. V.1. P.34.

2. Smallman R.E., Westmacott K.H. // Phil. Mag. 1957. V.2. P. 669.

3. Valiev R.Z., Alexandrov I.V. Nanostructured materials obtained by severe plastic deformation. Moscow, Logos, 2000. 272 p.  (Book in Russian).

X-ray diffraction measurements of the stacking fault probability combined with  TEM measurements of  the fault width results dislocation densities directly from the formula: alpha= rho x d111 W.  Where alpha stacing fault probability, rho  average dislocation density, d111 is the spacing of close-packed planes and W  is the average separation of partials. We found rho  9.7 x 1011 cm-2 for particle size of 20 micro, regardless the chemical composition. ' Met. Trans.  10A, 1505 (1979)', and Met. Trans.6A,493 (19759'.

By tracking up the annealing kinetics using  the stacking fault probability measurements at room temperature, one can also determine the chemical diffusivities in cold- worked (filings) Cu-10Zn (20 micron) and Cu*22.7Zn (75 micron) in alpha brasses.

 We found  1.7 x10-21 and 1.5 x 10-19 cm2 s-1,  respectively.

One can also estimate the athermal vacancy concentrations for these alloys and was found to  2.0 x 1019 and  3.0 x1019  cm-3 respectively.

Peak profile analysis should really be seen as qualitative for dislocation density measurements given the mathematical and practical limitations Article An evaluation of Diffraction Peak Profile Analysis (DPPA) me...

You can study papers, submitted by "T.Ungar" about the study of dislocation density by XRD.

You can find the method in my published article. It is for stainless steels but you can use it for other materials by finding the required constants.

Dr. Don Mikkola at Michigan Tech published many articles ~30 years ago on Fourier analysis of x-ray line broadening via XRD. Worked well with aluminum oxide. And, yes, you can get dislocations in Al2O3 at high strain rates.

The following article maybe helpful in the field:

http://www.scielo.br/pdf/mr/2015nahead/1516-1439-mr-1516-1439369214.pdf

δ=1/(Dp^2 ) using this equation you can find dislocation density. Where D is crystal size from Scherer’s formula..